Reflexivity of the space of transversal distributions

TL;DR

This paper proves that the space of smooth functions on the total space of a surjective submersion is a reflexive module over the space of compactly supported smooth functions on the base manifold, revealing a key functional-analytic property.

Contribution

It establishes the reflexivity of the space of smooth functions on a surjective submersion as a module over compactly supported functions, a novel structural insight.

Findings

The space of smooth functions on the total space is reflexive as a module.

Reflexivity holds in the context of surjective submersions.

The result links geometric structures with functional-analytic properties.

Abstract

We show that the Fr\'{e}chet space $\mathord{\mathcal{C}^{\infty}}(P)$ of smooth functions on the total space of a surjective submersion $\pi:P\to M$ is a reflexive $\mathord{\mathcal{C}^{\infty}_{c}}(M)$-module.